Example of real analytic function

Question

We were taught real analytic functions in class today. I am playing around trying to construct examples. I see exponential, sine, cosine and logarithmic functions (for $x > 0$). One function I am having trouble with is $f(x) = \frac{1}{1 + e^x}$. In spirit, this function is like $e^{-x}$, so I want to say it is real analytic, but not totally sure. Any help, please?

Answer 1

To recall that the reciprocal of an analytic function with no zeros is analytic is one way. For an argument see Is the reciprocal of an analytic function analytic?

Answer 2

Take any complex-analytic function $f=u+iv$. Then both $f,v$ are Real analytic. Or you can consider a function $f=(u(x,y), v(x,y))$ on $\mathbb R^2 $ and check that it satisfies Cauchy-Riemann: $u_x=v_y ; u_y =-v_x$.